Texturing deformation through a sequence of meshes
I solved the problem.  I did not use shape keys at all.

This can be done by copying the UV-mapping from the source mesh onto the target mesh (c.f. https://docs.blender.org/manual/en/2.79/editors/uv_image/uv/layout_management.html).  

Since both objects have the same topology, this is straightforward.